Fractional high order methods for the nonlinear fractional ordinary differential equation

摘要

In this paper, we consider the nonlinear fractional-order ordinary differential equation (0)D(t)(alpha)y(t) = f (y, t), (t > 0), n - 1 < alpha <= n, y((i)) (0) = Y-0((i)), i = 0, 1, 2, ..., n - 1, where f (y, t) satisfies the L-condition, i.e., vertical bar f (y(1), t) - f (y(2), t)vertical bar <= L vertical bar y(1) -y(2)vertical bar in t is an element of [0, T]. Fractional-order linear multiple step methods are introduced. The high order (2-6) approximations of the fractional-order ordinary differential equation with an initial value are proposed. The consistence, convergence and stability of the fractional high order methods are proved. Finally, some numerical examples are provided to show that the fractional high order methods for solving the fractional-order nonlinear ordinary differential equation are computationally efficient solution methods. (c) 2006 Elsevier Ltd. All rights reserved.